Counter-propagating parametric interaction with phonon-polaritons in periodically poled KTiOPO<sub>4</sub>

Hoon Jang; Anne-Lise Viotti; Gustav Strömqvist; Andrius Zukauskas; Carlota Canalias; Valdas Pasiskevicius

doi:10.1364/OE.25.002677

1. Introduction

KTiOPO₄ (KTP) and its isomorphs are well-established ferroelectric nonlinear optical materials used for second-order frequency conversion applications. Electric-field poling can be used in these crystals to tailor the spatial profile of the effective nonlinear coefficient to produce quasi-phase matched (QPM) structures. Similar to other ferroelectrics, e.g. LiNbO₃ and LiTaO₃, KTP is characterized by strong coupling of terahertz (THz) electromagnetic waves to the mechanical degrees of freedom, i.e. the transversal optical (TO) phonon modes, which gives rise to a very efficient phonon-polariton scattering [1–5]. The TO phonon modes involved in the polariton excitation are infrared-active with concomitant emergence of absorption for larger polariton wave-vectors. In ferroelectrics, most of these TO lattice vibrations are also Raman-active, although it is not a requirement for realizing polariton coupling [5,6]. The second-order nonlinearity, χ⁽²⁾, can have a large dispersion in the polariton frequency range [1,7–10]. Due to the coherent superposition of the non-resonant electronic contribution with the polar lattice vibrations, the overall χ⁽²⁾ can be enhanced (or reduced) in the spectral regions around the TO vibrational modes. In general, stimulated polariton scattering (SPS) involves both the second-order and third-order, Raman, nonlinearities [1,11,12]. As well as those previously reported in LiNbO₃ [13], efficient forward SPS (FSPS) with a small wave-vector of polaritons is behind the recent demonstrations of tunable THz nanosecond optical parametric oscillators in a single-domain KTP [14] and KTiOAsO₄ [15]. Coherence and relatively low absorption of small-k polariton waves generated in FSPS opened up possibilities for demonstration of polaritonic photonic crystals, waveguides and other intriguing opportunities for their use in on-chip THz applications [16–18].

The response times of the χ⁽²⁾ associated with the strongest TO phonons are of the order of 1 ps in KTP [4], which is much slower than the non-resonant electronic part of the nonlinearity. For instance, in THz generation employing optical rectification of near-infrared femtosecond pulses in QPM ferroelectrics [19–22], it is mostly the electronic part of χ⁽²⁾ that is harnessed due to the short excitation pulses. In such situation, coupling to other mechanical degrees of freedom in the lattice can also happen through an impulse generation of coherent phonons, which has been observed in various solid state materials [23,24]. It is tempting to think that, for pulses longer than the characteristic response time of the lattice dipoles, one could exploit the flexibility afforded by the QPM techniques and, at the same time, harness the enhanced χ⁽²⁾ nonlinearity offered by the lattice dipole system. However, we have recently demonstrated that FSPS with such long pump pulses is strongly suppressed in periodically-poled KTP (PPKTP) [12]. The FSPS suppression results from the interplay between the third-order Raman and the second-order nonlinearities associated with the same TO phonons, which results in the generation of antiphase polariton waves in adjacent ferroelectric domains and, therefore, leads to parametric de-amplification of the THz wave. Recent reports on THz generation in OPOs using periodically poled Mg:LiNbO₃ (PPLN) pumped with nanosecond pulses seem to show that, at least to some extent, similar suppression of the polariton generation takes place in this material as well [25,26].

In this work, we show that a rather surprising corollary of the FSPS suppression in PPKTP is a strong enhancement of backward SPS (BSPS). The observed BSPS is highly efficient with more than 50% of the pump energy being transferred to the backwards propagating Stokes. Moreover, as in backward Stimulated Raman scattering (BSRS), the Stokes pulses generated in BSPS also experience substantial compression in time domain. Superficially, BSPS bares certain similarities to BSRS, which was first observed in a CS₂ liquid in 1966 [27]. The signature of the BSRS as observed in long liquid cells (typically tens of cm long) is the generation of intense Stokes pulses in the backward direction with a pulse length much shorter than that of the pump [28]. However, BSRS requires relatively high intensities to become efficient, since forward scattering first takes place in most cases. Due to inherent symmetry in the stimulated Raman scattering, the backward and forward scattering cannot be disassociated, i.e., both processes happen at the same time. Although BSRS was proposed as a mechanism limiting beam radius in filaments under self-focusing conditions [29,30], and also as means for high-intensity ultrashort pulse generation in plasmas [31], the requirement for high pump intensity that is close to the optical damage threshold makes it rather difficult to utilize the backward stimulated Raman process in crystals [32]. In contrast to BSRS, however, the BSPS process in PPKTP can be well separated from the forward polariton scattering. We show here that the threshold and efficiency of the BSPS process in PPKTP is far below self-focusing or optical damage threshold. In fact, BSPS can be a strong competing process in counter-propagating QPM optical parametric devices [33,34].

2. Backward stimulated polariton scattering

The energy conservation, $ω_{s} = ω_{p} - ω_{p o l}$ , and the momentum conservation $k_{s} = k_{p} - k_{p o l}$ conditions are satisfied in SPS. Here, $ω$ is the cyclic frequency, $k$ is the wave-vector, and the indices, s, p, pol, denote Stokes, pump and polariton waves, respectively. The wave-vector diagrams for the FSPS and BSPS processes are shown in Fig. 1(a). Since polaritons are highly dispersive waves, the forward scattering generates the Stokes and the polariton waves non-collinearly. From the momentum conservation, it follows that the magnitude of the polariton wave-vector is given by:

| k_{p o l} | = \sqrt{{| k_{p} |}^{2} + {| k_{s} |}^{2} - 2 | k_{p} | | k_{s} | \cos φ_{s}}

where

φ_{s}

is the internal angle between the directions of the pump and the Stokes wave-vectors as shown in Fig. 1(a). For the pump propagating along the crystal x-axis and the polarization parallel to the crystal z-axis, i.e. in the X(ZZ)X configuration, the strongest polariton coupling arises from A1 symmetry TO phonons.

Fig. 1 (a) The phase-matching conditions for FSPS and BSPS. Only non-collinear FSPS is observed in single-domain KTP. (b) Polariton dispersion in KTP (black solid line) and absorption spectrum (blue line). Frequencies of TO (black dashed lines) and LO (red dashed lines) phonons used to model polariton dispersion and absorption. Red solid lines represent polariton wave-vectors calculated from Eq. (1) for the fixed pump wavelength of 807 nm and angles $φ_{s} = 0 °, 1.8 °, 3.4 °, 180 °$ . Polaritons observed experimentally in FSPS in single-domain KTP (blue dots) [12], and BSPS in PPKTP (green dots) with e.g. 36 µm periodicity.

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Figure 1(b) compares the phase-matched FSPS and BSPS marked on the polariton dispersion curve and the corresponding absorption spectrum. The black solid line shows the polariton dispersion calculated by taking into account ten strongest TO phonon lines in KTP, where we used the Lorentzian series representation of the complex dielectric function with the phonon data from [35]. The dashed black and red lines represent the central frequencies of the TO and the associated LO phonons, whereas the solid blue line represents the calculated absorption spectra owing to these infrared-active TO phonons. In the vicinity of these phonon resonances, the polariton dispersion strongly dictates the noncollinear phase-matching configuration of FSPS in a single-domain KTP [12,36]. The blue dots in Fig. 1(b) represent the experimental data that corresponds to the polaritons generated in the FSPS processes at the pump wavelength of 807 nm. The red lines in this figure have been calculated from Eq. (1) using the fixed wave-vector $| k_{p} | = 1.44 \times 10^{3}$ cm⁻¹ at 807 nm and the angle of $φ_{s} = {1.8}^{\circ}, {3.4}^{\circ}$ , corresponding to the experimentally measured internal Stokes angles in KTP. The intersection of these curves with the polariton dispersion gives the expected phase matching points in good agreement with the experiment.

In the case of BSPS, the Stokes wave is generated in opposite direction to the pump, $φ_{s} = 180^{\circ}$ , as described in Fig. 1(a). Therefore, the momentum conservation, $| k_{p o l} | = | k_{p} | + | k_{s} |$ , requires a large polariton wave-vector, i.e. $| k_{p o l} | \approx 2 | k_{p} |$ . The required $| k_{p o l} |$ as a function of frequency in the BSPS process was calculated from Eq. (1) and shown by the red solid line marked $φ_{s} = 180^{\circ}$ in Fig. 1(b). The green data points mark the polariton frequencies observed in our experiment. In this range of large momenta, the polariton dispersion is rather flat and asymptotically approaching to a TO phonon frequency responsible for that particular polariton branch till the Brillouin zone boundary. In this case, the refractive index of the polaritons generated in BSPS can be very large, namely:

n_{p o l} = (n_{p} + n_{s}) \frac{ω_{p}}{ω_{p o l}} - n_{s}

where n_s, n_p are refractive indices of the Stokes and the pump, respectively. The values of the indices of refraction and other parameters for the polaritons at the green data points in Fig. 1(b) are given in the Table 1.

Table 1. Parameters of the polaritons observed in backward stimulated polariton scattering (BSPS) in PPKTP with e.g. 36 µm periodicity.

View Table

Please note that the phase-matching condition for BSPS in PPKTP does not involve the grating vector in the equation. This is because the slow polariton waves do not experience the poling periods before the absorption takes over. Large absorption of the polaritons would limit the effective interaction length to about 1 µm for the strongest polariton lines in BSPS (8 THz and 20.6 THz), which corresponds to only few polariton wavelengths in the crystal. This also renders the above-mentioned suppression mechanism of polariton scattering inefficient in PPKTP in the backward-propagating geometry. Instead, as presented below, it is observed that BSPS gets strongly enhanced in PPKTP with the phase-matching condition independent of the poling periods.

3. Experimental results and discussion

In order to study the effect of $χ^{(2)}$ structuring on the BSPS threshold, we compared single-domain KTP crystals with several PPKTP structures with different periodicities ranging from 9 µm to 500 µm. All the crystals had the same crystal length of 11.6 mm, which was fully covered by each grating. As a pump source, we used a picosecond Ti:Sapphire regenerative amplifier operating at 807 nm with a pulse spectral bandwidth of 0.7 nm to produce the pulse length tunable from 1.5 ps to about 200 ps by adjusting dispersion in the amplifier. BSPS was observed in the $X (Z Z) \bar{X}$ configuration, i.e. with the pump beam propagating along the crystal x-axis and polarization parallel to the crystal z-axis. The PPKTP structures were slightly tilted to prevent Fresnel reflections from interfering with the measurements. In this configuration, it was verified that the BSPS Stokes beam is generated in the counter-propagating direction to the pump. Figure 2(a) shows the measured BSPS threshold as a function of poling period in PPKTP. As the poling period becomes smaller, FSPS is increasingly suppressed [12], eventually leaving BSPS the dominant polariton scattering process in PPKTP with the 9 µm period. In the PPKTP crystals with long periodicities, the BSPS threshold approaches that of the single-domain KTP. In these crystals, the BSPS has substantially higher pump threshold compared to the FSPS, consequently making the BSPS process inefficient. Two dominant polariton lines, 8 THz and 20.6 THz, have been observed in BSPS for all crystals, which is shown in Fig. 2(b). The frequency of the polaritons giving rise to BSPS did not depend on the poling period. For the crystals with periodicities below 36 µm, where BSPS is already dominant, a new and weaker polariton line at 5.5 THz starts to emerge. We tentatively attribute it to 6.1 THz TO phonon, which has relatively low oscillator strength [35] compared with much stronger vibrations at 8.04 THz and 21 THz.

Fig. 2 (a) The observed intensity-thresholds of BSPS as a function of poling period with a fixed pump pulse duration of 47 ps. The error bars stem primarily from the uncertainty in beam-size measurements. (b) The pump at 807 nm and the observed BSPS spectrum associated with the TO phonon lines in PPKTP with the poling period of 36 µm. The amplitudes of the 5.5 THz and 8 THz are suppressed with respect to 21 THz line due to lower reflectivity of the beam-splitter in front of the PPKTP crystal. The Fresnel reflection of the pump propagates in a slightly different direction owing to intentional tilting of the crystal.

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According to Fermi’s golden rule, the SPS coupling efficiency associated with a particular TO phonon mode would depend on the density of polariton states. The spectrum of the density of states (DOS) can be estimated from polariton dispersion via the Hilbert transform of ${(2 π k_{p o l})}^{2} \partial k_{p o l} / \partial ω$ . The ratio of the normalized DOS to the normalized absorption at the phase-matched polariton frequency can be used as a “figure of merit” that indicates the expected net gain for each polariton mode. These ratios for the polaritons responsible for BSPS at 5.5 THz, 8 THz and 20.06 THz are respectively 0.508, 1.6, and 0.823. Therefore, it is expected that BSPS involving 8 THz polaritons would have the lowest threshold, as indeed is confirmed in our experiments with different PPKTP samples.

We emphasize that the BSPS process is very efficient in PPKTP structures where FSPS is totally suppressed. As shown in Fig. 3(a), the efficiency of BSPS reaches 70% at the pump intensity ~3 times the threshold. Here, the PPKTP structure with 9 µm periodicity was pumped with 200 ps-long pulses. In such situations, there is no quasi-phase matching of any forward-propagating parametric processes, which makes BSPS extremely efficient with full power transfer from the pump. This is in total contrast to BSRS, the third order process, where the backward scattering is relatively inefficient due to dominant forward scattering processes. This behavior of the BSPS process bears similarities with counter-propagating three-wave mixing through χ⁽²⁾ nonlinearity that gives rise to mirrorless optical parametric oscillation (MOPO) [33,37]. The oscillation threshold in MOPO is inversely proportional to the square of the pump pulse length, $τ^{- 2}$ , provided that the pulses have the effective interaction length $L = τ c / (2 n_{p})$ shorter than that of the crystal. Indeed, the fit to the experimental data in Fig. 3(b) shows that BSPS threshold also depends on the inverse power function on the pump pulse length, although it scales as $τ^{- 1.62}$ and $τ^{- 1.73}$ for the BSPS at 8 THz and 20.6 THz, respectively. In contrast to BSPS and MOPO, the gain in BSRS is proportional to ${(Γ_{T O} + Γ_{L})}^{- 1}$ [38], where $Γ_{T O}$ is the spontaneous Raman bandwidth and $Γ_{L}$ is the laser pulse bandwidth. Therefore, the gain in BSRS should be largely independent on the laser pulse length $τ$ , provided that $Γ_{T O} τ > > 1$ . The bandwidths of TO modes at 8.1 THz and 21 THz would support the transform-limited Gaussian pulses of 2 ps and 0.8 ps, respectively, which are substantially shorter than the pump pulses used in our experiments.

Fig. 3 (a) Pump depletion and the corresponding BSPS efficiency as a function of pump intensity with a fixed pump pulse duration of 200 ps in PPKTP with the periodicity of 9 µm. (b) The measured dependence of the BSPS threshold on the pump pulse length for the Stokes at 8 THz (red squares) and 20.6 THz (blue squares). Blue and red lines represent power fits to the measured data. The black solid line represents inverse square dependence, which is expected in lossless counter-propagating parametric oscillator. The PPKTP structure with the periodicity of 9 µm was used in these measurements.

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One of the signatures of the BSRS is the generation of intense Stokes spikes which are substantially shorter than the pump pulses [27,28]. In our BSPS experiment with PPKTP, the indication of a high-peak intensity in the Stokes field initially came from the observation that the non-phase matched second harmonic generation in the backward direction was markedly higher than that produced by the forward propagating pump. We subsequently measured the pulse length of the backward Stokes via cross-correlation with a small part of the pump pulse that was compressed to the transform-limited length of 1.5 ps. The Stokes and the compressed reference pump after passing through a variable delay line were superimposed non-collinearly in a type-I phase-matched BBO crystal. The resulting sum-frequency generation was imaged through a diffraction grating onto a Si p-i-n photodetector, where the grating enabled separation of the sum-frequency fields stemming from the Stokes at 8 THz and 21 THz. The cross-correlation traces of the backward Stokes at different pump intensities are shown in Fig. 4(a), while that of the incoming pump is displayed in Fig. 4(b). The pulse shape of the backward propagating wave depended strongly on the pump intensity above the threshold. The general tendencies can be observed in the traces of the 8 THz Stokes. Just above the threshold, the pulse is a well-defined Gaussian with the FWHM of about 19 ps. As the pump intensity is increased up to 4.7 GW/cm², the backward Stokes pulse acquires a double-peak structure which could be fit with a sum of two Gaussian pulses, one centered at 47 ps with the FWHM of 41 ps, and the other at 82 ps with the FWHM of 9 ps. Similar structure, albeit more compressed in time, is retained at the highest pump intensity of 11 GW/cm². The backward Stokes pulse corresponding to the 20.6 THz polaritons always had a single-Gaussian shape, which was centered at 45 ps with the FWHM of 22 ps at the highest pump intensity.

Fig. 4 Cross-correlation measurements for backward propagating Stokes (a) and the pump (b). In (a), squares denote Stokes corresponding to 8 THz polaritons at different pump intensities (solid black: 2.2 GW/cm², solid blue: 4.7 GW/cm², open black: 11 GW/cm²), while red circles denote Stokes corresponding to 21 THz polaritons at the pump intensity of 11 GW/cm². Red solid lines represent Gaussian fits.

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The change in the pulse shape indicates that the BSPS is not a stationary process, which differs from the near-infrared counter-propagating parametric oscillators. This difference is ascribed to the fact that the absorption of the polariton wave generated in BSPS is very large, as shown in Table 1. The role of the large polariton absorption in the counter-propagating geometry is shown in Fig. 5 by calculating the normalized intensity distributions of polaritons and Stokes along the crystal. The results were obtained by numerically solving coupled-wave equations with second-order nonlinear interaction of monochromatic waves in counter-propagating geometry [39]. As the absorption of the polariton wave is increased, the parametric gain region is increasingly concentrated at the beginning of the crystal. The intensity buildup in Stokes field, however, is not achieved immediately. One can estimate the time-scale of the Stokes intensity buildup as $L_{c} / (v_{g p} + v_{g s})$ , where L_c is the crystal length, and v_gp, v_gs are the group velocities of the pump and the signal, respectively. For the crystals used in our experiment, this time scale is about 40 ps, which generally agrees with the position of the first intensity peaks in the Stokes cross-correlation traces. The Stokes and polariton buildup strongly depletes downstream pump, which leads to a self-termination of the process as the first Stokes pulse exits the crystal. At higher pump intensities, however, the buildup process can repeat itself, but on a shorter temporal scale, as some parametric gain is still present at the beginning of the crystal. This second pulse then should be shorter and more intense. This qualitative picture explains the experimental observations. It should be noted that BSRS would also result in multiple pulsing if the length of the gain medium is shorter than the optical length of the pump pulse. However, the Stokes pulse in BSRS should be strongly asymmetric with the leading edge of the pulse being much steeper, since the backward Stokes pulse continuously encounters the undepleted part of the pump as it sweeps through it [28].

Fig. 5 Calculated normalized stationary signal (Stokes, blue lines) and idler (polariton, red lines) intensity distributions for different values of polariton absorption coefficient. The arrows indicate the directions of propagation for each wave. Pump intensity of 13 GW/cm² is used in the calculations with the effective nonlinear coefficient of 183 pm/V [12]. The intensity distributions give increasingly shorter amplification lengths as the absorption increases. Note that the absorption coefficient of 40 cm⁻¹ already limits the parametric gain region at the beginning of the crystal. The actual polariton absorption coefficient involved in BSPS is as large as 10 ⁴ cm⁻¹ as previously shown in Table 1.

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As already shown, the BSPS process can become a very efficient and dominant one in PPKTP. The process can be used in a number of applications where a high peak intensity or a frequency-shifted beam in the direction conjugate to the pump is required. On the other hand, in circumstances with near- and mid-infrared parametric QPM devices, the BSPS might be detrimental. In a good-quality PPKTP structures where other processes are phase-matched, the BSPS is usually not observed due to the pump depletion before the BSPS can reach its threshold. However, fabrication of QPM structures can impart errors to the periodicity in the structures, or to the duty cycle of ferroelectric domains. Therefore, it is instructive to estimate the error margin above which the BSPS is expected to become a competing nonlinear process. This is especially important for QPM structures with sub-micrometer periodicity, which is used in mirrorless OPOs and other counter-propagating parametric frequency converters. The product of the parametric gain and the interaction length $g L$ has the same functional form for the BSPS and the near-infrared parametric devices:

{(g L)}^{2} \propto \frac{d_{e f f}^{2} L^{2}}{n_{s} n_{i} λ_{s} λ_{i}}

where d_eff, n_s, n_i, λ_s, λ_i, are, respectively, the effective nonlinear coefficient, the refractive indices of the signal (Stokes in BSPS) and the idler (polariton in BSPS), the wavelengths of the signal and the idler. The nonlinear coupling for BSPS can be quite large, owing to the large effective nonlinear coefficient in the vicinity of polariton resonances [12]. Taking a specific example of the PPKTP MOPO pumped at 800 nm, generating the signal, λ_s = 1.18 µm, and the idler λ_i = 2.9 µm, we get

{(g_{N I R} L_{N I R})}^{2} \propto 8.9 L_{N I R}^{2} [p m^{2} / V^{2}]

. Here, L_NIR is measured in microns, and represents the effective length of the QPM structure, which might be different from the physical length of the QPM structure due to fabrication errors. For the BSPS process, we have

{(g_{p o l} L)}^{2} \propto 3.5 L^{2} [p m^{2} / V^{2}]

where

L

is now the physical length of the QPM structure. As clarified above, the slow polariton wave will not move appreciably before being absorbed. Therefore, the errors in the domain periodicity are irrelevant to the efficiency of the BSPS process. From this estimate, it is clear that the near- or mid-infrared parametric QPM process in PPKTP will always have lower threshold than that of BSPS in a perfect QPM structure with no errors. However, both processes will have equal gain if the effective QPM length is 38% shorter than the physical length of the structure, provided that the QPM errors are uniformly distributed over the physical structure. For example, if we consider the asymmetry in the duty cycle, which is one of the most common fabrication errors encountered in sub-micrometer periodicity structures, this error will reduce the effective nonlinearity d_eff for the near-infrared parametric process [40]. In this case, the BSPS will have equal gain with the near-infrared parametric process when the duty cycle is reduced from the perfect 0.5 to about 0.22. For such imperfect structures, there is a high probability that the BSPS will become efficient and prevent counter-propagating parametric oscillation in near- and mid-infrared.

4. Conclusions

In conclusion, we have shown a rather surprising dominance of the BSPS process in PPKTP structures, where it was possible to convert 70% of pump power into Stokes propagating in the backward direction. The suppression of the forward polariton scattering in the χ⁽²⁾ structures with periodicity smaller than about 36 µm makes the BSPS very efficient, owing to the strong enhancement of the effective second-order nonlinearity near the phonon-polariton resonances. The enhancement is most prominent for the pump pulses longer than the inverse bandwidth of the TO-phonon modes that are responsible for the polariton branches. BSPS can be understood primarily as a second-order parametric process in counter-propagating geometry, although this does not preclude a role of third-order Raman scattering in the same crystals. The BSPS is essentially a non-stationary process where the backwards generated Stokes is comprised of pulses with the shape dependent on the pump intensity. This includes a substantial pulse compression with higher peak intensity compared with the pump. There are certain superficial similarities between BSPS and BSRS, which can be surprising if one considers that SRS is produced by localized dispersionless lattice optical phonons whose phase is essentially irrelevant for the SRS outcome. In general, the relative phases of the interacting waves in the second-order parametric process play an important role in the phase-matching condition. In BSPS, however, the polariton wave is extremely slow and strongly absorbed such that the phase information cannot be transferred so far. Therefore, the polariton phase will be dictated by the interaction of the counter-propagating pump and Stokes. In principle, this physical mechanism is rather general and should not be limited to KTP and its isomorphs.

Acknowledgments

We would like to acknowledge the Swedish Research Council (VR), Linné Center ADOPT, and Göran Gustafsson Foundation, and Swedish Foundation for Strategic Research for financial support of this work.

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f_pol [THz]	λ_pol [nm]	n_pol	k_pol [10⁴ cm⁻¹]	1/α [µm]
5.5	220	247	28.5	47
8	221	169	28.4	0.5
20.6	225	65	27.9	1

Counter-propagating parametric interaction with phonon-polaritons in periodically poled KTiOPO₄

Abstract

1. Introduction

2. Backward stimulated polariton scattering

3. Experimental results and discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (3)

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